Once you know the one-digit cubes by heart, look at the table again. In the table, the last digit of each of the cubes have been highlighted. Notice that each cube ends in a unique digit, not shared by any of the other cubes. This will help simplify the full process later.
Also note that most of the digits, when cubed, end in the same digit (for example, 9^3^ ends in a 9, 5^3^ ends in 5, etc.). The only exceptions are 2, 3, 7 and 8. These are easily remembered, too. 2^3^ "ends" in 8, and 8^3^ ends in a 2. Similarly, 3^3^ ends in 7 and 7^3^ ends in 3.
Once you know the one-digit cubes and their roots, and can remember which ending digit in a cube refers to which digit of its cube root, you're ready to move on to the next section.
I teach the process of finding cube roots with an example cube, 148,877. First, consider the numbers on either side of the comma as two separate numbers. In this case, think of the answer as 148 and 877.
First, take the numbers to the left of the comma, 148. Figure out which cube is the closest one to this number, without going over it. With 148, 125 (5^3^), is the closest cube root to 148, without going over it. This means that the tens digit of the answer is 5. We already know that the answer is somewhere in the 50's.
Getting the final digit is even simpler. Looking at the numbers to the right of the comma, we have 877. All you do here is look at the rightmost digit of the number (7, in this case). Do you remember which number's cube ends in 7? It's 3.
This means that the cube root of 148,877 is 53. If you check on a calculator, you'll find it is indeed 53.
Remember, regardless of the length of the two-digit cube, the dividing point will always be the comma. If you're given 4,913, for example, you break it up into 4 and 913. The closest cube to 4 without going over is 1 (which has a cube root of 1). The 3 in 913 tells us that the ones digit in the cube root is a 7, so the cube root of 4,913 is 17.